Modelling of the B-type binaries CW Cep and U Oph: A critical view on dynamical masses, core boundary mixing, and core mass
AAstronomy & Astrophysics manuscript no. johnston_accepted_v1 c (cid:13)
ESO 2019May 30, 2019
Modelling of the B-type binaries CW Cep and U Oph
A critical view on dynamical masses, core boundary mixing, and core mass
C. Johnston , K. Pavlovski , and A. Tkachenko Instituut voor Sterrenkunde, KU Leuven, Celestijnenlaan 200D, 3001, Leuven, Belgium, e-mail: [email protected] Department of Physics, Faculty of Science, University of Zagreb, Bijeniˇcka cesta 32, 10000 Zagreb, CroatiaReceived Date Month Year / Accepted Date Month Year
ABSTRACT
Context.
Intermediate-Mass stars are often overlooked as they are not supernova progenitors but still host convective cores andcomplex atmospheres which require computationally expensive treatment. Due to this, there is a general lack of such stars modelledby state of the art stellar structure and evolution codes.
Aims.
This paper aims to use high-quality spectroscopy to update the dynamically obtained stellar parameters and produce a newevolutionary assessment of the bright B0.5 + B0.5 and B5V + B5V binary systems CW Cep and U Oph.
Methods.
We use new spectroscopy obtained with the Hermes spectrograph to revisit the photometric binary solution of the twosystems. The updated mass ratio and e ff ective temperatures are incorporated to obtain new dynamical masses for the primary andsecondary. With these, we perform isochrone-cloud based evolutionary modelling to investigate the core properties of these stars. Results.
We report the first abundances for CW Cep and U Oph as well as report an updated dynamical solution for both systems. Wefind that we cannot uniquely constrain the amount of core boundary mixing in any of the stars we consider. Instead, we report theircore masses and compare our results to previous studies.
Conclusions.
We find that the per-cent level precision on fundamental stellar quantities are accompanied with core mass estimates tobetween ∼ − ff erences in analysis techniques can lead to substantially di ff erent evolutionary modelling results,which calls for the compilation of a homogeneously analysed sample to draw inference on internal physical processes. Key words. stars: keyword 1 – stars: keyword 2 – stars: keyword 3
1. Introduction
The model independent estimates of the absolute dimensionsof and distances to stars provided by eclipsing binary systemsserve as a fundamental calibrator in modern astrophysics. In thebest cases, such systems o ff er dynamical mass and radius esti-mates better than one per-cent (Torres et al. 2010). Such pre-cise measurements combined with the powerful constrains ofco-evolution and identical initial chemical composition have al-lowed the thorough investigation of the importance of rotation instellar evolutionary theory (Brott et al. 2011a,b; Ekström et al.2012; de Mink et al. 2013; Schneider et al. 2014; Ekström et al.2018), the calibration of pre through post main-sequence evolu-tion (Torres et al. 2013; Higl & Weiss 2017; Beck et al. 2018b;Kirkby-Kent et al. 2018), the critical investigation of magneticfields in stars (Takata et al. 2012; Grunhut et al. 2013; Torreset al. 2014b; Pablo et al. 2015; Kochukhov et al. 2018; Wadeet al. 2019), the calibration of distances (Guinan et al. 1998;Ribas et al. 2000a, 2005; Hensberge et al. 2000; Bonanos et al.2006; Pietrzy´nski et al. 2013; Gallenne et al. 2016; Suchomskaet al. 2019), the investigation of abundances and rotational ve-locities (Pavlovski & Hensberge 2005; Pavlovski & Southworth2009; Pavlovski et al. 2009, 2018; Simón-Díaz et al. 2017),as well as the calibration of asteroseismic modelling (De Catet al. 2000, 2004; Aerts & Harmanec 2004; Schmid et al. 2015;Schmid & Aerts 2016; Beck et al. 2018a,b; Johnston et al. 2019).Additionally, the advent of such precise measurements has led tothe uncovering of the reported systematic discrepancy between masses obtained via dynamics or empirical spectral relations andfitting theoretically calculated evolutionary tracks (Herrero et al.1992; Ribas et al. 2000b; Tkachenko et al. 2014). This discrep-ancy has served as the centrepiece of intense debate over the im-portance of convective core boundary mixing in stellar evolutiontheory (Ribas et al. 2000b; Torres et al. 2010, 2014a; Tkachenkoet al. 2014; Stancli ff e et al. 2015; Claret & Torres 2018; Con-stantino & Bara ff e 2018; Johnston et al. 2019).In general, both element and angular momentum transportprocesses throughout a star are poorly calibrated (Aerts et al.2019). It is a well known short-coming of most 1-D theoret-ical descriptions of convection that convective boundaries arenot well described (Hirschi et al. 2014). Proposed as a meansto remedy this short-coming, the inclusion of convective coreovershooting as a way to increase near-core mixing in evolu-tionary models is now highly debated, with several competingstudies claiming that models with and without overshooting canreproduce observed binaries across di ff erent mass ranges andevolutionary stages (Andersen et al. 1990; Schroder et al. 1997;Pols et al. 1997; Claret 2007; Stancli ff e et al. 2015; Claret &Torres 2016, 2017; Higl & Weiss 2017; Constantino & Bara ff e2018). Convective core overshooting is a phenomenon theoreti-cally predicted in intermediate- to high-mass stars with a convec-tive core where the inertia of a convectively accelerated mass el-ement propels said mass element beyond the convective bound-ary described by the Schwarzchild stability criterion into thestably stratified radiative region (Zahn 1977; Roxburgh 1978; Article number, page 1 of 17 a r X i v : . [ a s t r o - ph . S R ] M a y & A proofs: manuscript no. johnston_accepted_v1
Zahn 1991; Maeder 2009). The mathematical form of the im-plementation into stellar evolutionary codes is not universallyagreed upon, with di ff erent forms having been successfully usedto describe both binary (Ribas et al. 2000b; Guinan et al. 2000;Tkachenko et al. 2014; Claret & Torres 2016, 2017) and astero-seismic observations (Briquet et al. 2007; Moravveji et al. 2015,2016; Van Reeth et al. 2016; Johnston et al. 2019). To date, twosuch descriptions have been implemented in 1-D stellar evolu-tion codes: i) convective penetration where the temperature gra-dient in the overshoot region is the adiabatic one, ∇ T = ∇ ad ,and ii) di ff usive overshooting where the temperature gradient isthe radiative one, ∇ T = ∇ rad . This di ff erence results in a fullychemically and thermally mixed extended region in the case ofpenetration, e ff ectively meaning the core is extended and thusmore massive. In the case of di ff usive overshooting, the extendedregion is only partially chemically mixed, and any increase incore mass is due to the transport of chemicals into the convec-tive core via this chemical mixing. In either case, the convectivecore will thus have more hydrogen available to burn (or He in theHe-core burning phase), thus extending the main-sequence (MS)lifetime of the star, having a pronounced e ff ect on the morphol-ogy of evolutionary tracks. Alternatively, some studies have usednear-core rotational mixing to enhance the core mass, e ff ectivelyproducing the same situation where more rotational mixing cor-responds to a more massive core. We point out that in 1D di ff u-sive codes, the implementations of overshooting and rotationalmixing are seemingly di ff erent but both are able to function asa proxy for the total amount of near-core mixing, whatever thephysical cause, and that both prescriptions contain un-calibratedparameters. We adopt the approach of using overshooting as ageneral proxy for the total amount of near-core mixing, whateverits physical cause (rotation, convection, magnetism, waves). Themass-discrepancy reported between either spectroscopic (Her-rero et al. 1992) or dynamical masses (Guinan et al. 2000; Ribaset al. 2000b; Claret 2007; Tkachenko et al. 2014) and evolu-tionary masses has traditionally been resolved by increasing theamount of overshooting in a stellar model. This increase in over-shooting e ff ectively increases the core mass at a given age, mim-icking a more massive star.It was theoretically outlined that the extent of an overshoot-ing region would be limited by the total energy (mass) of the core(Roxburgh 1992), and hence the mass of the star. This theoreti-cal prediction has been investigated by numerous studies, someclaiming no significant mass dependence (Schroder et al. 1997;Pols et al. 1997; Stancli ff e et al. 2015; Constantino & Bara ff e2018), while others claim a statistically significant mass depen-dence (Claret 2007; Claret & Torres 2016, 2017, 2018, 2019).Yet another body of work suggests caution at the ability to con-strain overshooting from classical observable quantities giventhe sensitivity of the data and methodologies (Valle et al. 2016;Higl & Weiss 2017; Valle et al. 2017, 2018; Johnston et al. 2019;Constantino & Bara ff e 2018). On the theoretical side, Valle et al.(2016) studied the ability for models to uniquely describe a setof observables, revealing an inability to uniquely constrain over-shooting. This result was supported by the findings of Valle et al.(2018) and Constantino & Bara ff e (2018) who show that tradi-tional observed quantities do not provide enough discriminatingpower to uniquely constrain overshooting, with Constantino &Bara ff e (2018) being unable to reproduce the mass-dependenceof overshooting reported by Claret & Torres (2016). Further-more, Johnston et al. (2019) showed that even with the inclusionof asteroseismic information, the extent of overshooting, stel-lar mass, and age cannot be uniquely constrained when properlyaccounting for correlated nature of stellar model parameters. In- stead, Johnston et al. (2019) suggest that the mass and radius ofthe convective core should be reported and considered in placeof the overshooting.In this paper, we follow the paradigm of Johnston et al.(2019) to investigate the ability of well detached double-linedeclipsing binaries (EBs) to probe the core mass. Additionally,we investigate the comparatively sparsely sampled mass rangeof 4-6 M (cid:12) , connecting the mass ranges intensively covered byClaret & Torres (2016, 2017) and Pols et al. (1997); Higl &Weiss (2017). We revisit the intermediate- to high-mass double-lined EBs CW Cep and U Oph with new spectroscopy and radialvelocities to obtain updated mass and radii estimates. In Sec-tion 2, we will provide an overview of both systems, includingpast modelling e ff orts. In Sections 3 and 4 we discuss the newspectroscopy, the newly determined orbital elements from spec-tral disentangling, and the determination of spectroscopic quan-tities from the disentangled spectra, respectively. Section 5 de-tails the modelling procedure and results for both systems withthe mass ratio fixed as derived in the previous section. Section 6covers our evolutionary modelling procedure. Sections 6.2 and 7discuss the newly determined mass and radii estimates for eachsystem, the modelling results, and places them in the context ofthe larger modelling e ff orts of the community. Following the re-sults of Constantino & Bara ff e (2018) and Johnston et al. (2019),we report and discuss the estimated core mass and overshootingfrom our modelling procedure.
2. Literature overview on CW Cep and U Oph
The detached double-lined EB CW Cephei (HD 218066, V = . M = . − .
49 M (cid:12) and M = . − .
05 M (cid:12) (Popper 1974, 1980;Clausen & Gimenez 1991; Han et al. 2002), placing this sys-tem at the lower end of the high-mass sequence. This spread inmasses results in an uncertainty of ∼
13% compared to the me-dian value (solution b by Han et al. (2002)). The quality of thephotometric light-curve solution, in particular the determinationof the masses and radii, has been restricted by uncertainty in themass and light ratio, respectively. This problem has been exten-sively discussed by Clausen & Gimenez (1991),who found thatthe spread in ratio of radii is also accompanied by a significantspread in the sum of the radii. Subsequent analysis of their ownnew photometry by Han et al. (2002) and Erdem et al. (2004)did not settle issue as they used a di ff erent methodology fromClausen & Gimenez (1991). Namely, Han et al. (2002) did notprefer the photographically determined light ratio over that re-turned from the lightcurve modelling, and Erdem et al. (2004)allowed for the possibility of a-synchronous rotation in the com-ponents, which alters the light ratios derived from photometricmodelling. Comparing the radii derived by di ff erent previousanalyses (a complete set of the references are given in Han et al.(2002)), a spread of ∼
8% is found.Apsidal motion was detected in CW Cep by Nha (1975) withimprovements to the apsidal period made by Han et al. (2002),Erdem et al. (2004), and Wolf et al. (2006). The last authors set-tled the apsidal period to U = . ± . e = . Article number, page 2 of 17. Johnston et al.: Modelling of the B-type binaries CW Cep and U Oph from the NASA TESS mission (Ricker et al. 2015) promise toprovide hitherto unseen constraints on the apsidal motion ob-served in this system.Due to a distinct lack of constraints on the metallicity ofCW Cep, the unique determination of evolutionary models forCW Cep has proven di ffi cult (Clausen & Gimenez 1991). Todate, several age estimates for CW Cep exist, with Clausen &Gimenez (1991) reporting an age of τ = (10 ±
1) Myr, placingboth components in the first half of the main-sequence. In theirfitting work, Ribas et al. (2000c) derived a much younger sys-tem with τ = . ± . τ = . ± . Z = . ± . Y = . ± . ∼ v sin i =
520 km s − for both components. It should benoted that Schneider et al. (2014) used a less massive solution intheir modelling than Ribas et al. (2000c) by ∼ . (cid:12) for the pri-mary and ∼ . (cid:12) for the secondary and fixed the metallicity oftheir tracks to solar. Furthermore, Blaauw et al. (1959) identifiedCW Cep to be a member of the Cep OB3, one of the smaller asso-ciations in the Orion arm. Blaauw (1961) also indicated that thisassociation is composed of two subgroups. CW Cep is located inthe older subgroup for which Clausen & Gimenez (1991) foundan average age of about 10 Myr in perfect agreement with theage they obtained for CW Cep. However, in a comprehensivestudy of a new homogeneous U BVRI photometry and member-ship Jordi et al. (1996) obtained ages of 5.5 and 7.5 Myr for thetwo subgroups, in disagreement with the ages derived by bothClausen & Gimenez (1991) and Schneider et al. (2014).CW Cep is also characterised as an intrinsically variable po-larized object (Elias et al. 2008). Both CW Cep and another earlyB + B binary system AH Cep, were observed with the ChandraX-ray Telescope in search for evidence of a wind-wind colli-sion (Ignace et al. 2017). Although CW Cep and AH Cep arecomprised of stars with similar properties (c.f. Pavlovski et al.(2018)), X-rays were only detected for AH Cep, despite it be-ing nearly a factor 2 further away than CW Cep. The authorscould not disentangle, however, whether the X-rays detectedfrom AH Cep were caused by colliding winds, or perhaps frommagnetic activity originating in one of the other components ofthe quadruple system of AH Cep (Ignace et al. 2017).
U Oph (HD 156247, V = .
92 mag) is a detached double-linedEB comprised of two B5V components. Much like in the caseof CW Cep, the dynamical solution of U Oph su ff ers from un-certainties in the light and mass ratios from spectral analysis.The reported masses for U Oph span from M = . − .
27 M (cid:12) and M = . − .
78 M (cid:12) , whereas the reported radii span from R = . − .
48 R (cid:12) and R = . − .
11 R (cid:12) (Holmgren et al.1991; Vaz et al. 2007; Wolf et al. 2006; Budding et al. 2009).This represents an uncertainty of ∼
7% and ∼
5% in M and M and an uncertainty of ∼
6% and ∼
3% in R and R whencompared to the most recent solution by Budding et al. (2009).Additionally, a wide range of e ff ective temperatures has beenreported for both components, with di ff erences up to 3 000 K (Clements & Ne ff ff er & Kopal(1951), the OAO-2 spacecraft photometery of Eaton & Ward(1973), or both. However, the work of Vaz et al. (2007) andBudding et al. (2009) relies on new photometric and spectro-scopic data. While all of these analyses used di ff erent modellingmethodologies, codes, and assumptions, they produce derivedquantities within a rather small range, as discussed above, andwith high precision, which is promising. U Oph displays a veryrapid apsidal motion with a period of U ≈
20 yr attributed toa distant third body (Koch & Koegler 1977; Kaemper 1986;Wolf et al. 2002). The apsidal motion has been studied thor-ougly with several proposed apsidal periods, some as large as 55yr (Frieboes-Conde & Herczeg 1973; Panchatsaram 1981; Wolfet al. 2002; Vaz et al. 2007). Several recent studies have tried toconstrain the nature of the tertiary component, reporting an or-bital period of P ≈
38 yr and M ≈ (cid:12) (Kaemper 1986; Wolfet al. 2002; Vaz et al. 2007; Budding et al. 2009).Largely due to uncertainties in its metallicity, there havebeen several discrepant ages reported for U Oph. Holmgren et al.(1991) first reported an age of ∼
40 Myr for U Oph when com-pared to evolutionary tracks without overshooting, and an age ∼
63 Myr when compared to evolutionary tracks with overshoot-ing. Later, Vaz et al. (2007) compare their solution to evolution-ary tracks of di ff erent metallicities, considering the apsidal con-stant as an additional constraint in their modelling and find thebest agreement with isochrones for ∼
40 Myr, ∼
52 Myr, and ∼
62 Myr calculated at Z = . , . .
01, respectively.Budding et al. (2009) perform their own evolutionary analysis,again with di ff erent codes and solutions compared to the pre-vious evolutionary modelling attempts, and arrive at an averageage estimate of ∼
38 Myr between the two components for trackscalculated at Z = .
02. The authors also note that a younger so-lution is found at ∼
30 Myr from tracks calculated at Z = . = ∼
41 Myr forthe system. Budding et al. (2009) provide a comprehensive anddetailed discussion of U Oph, to which we refer the reader foradditional information.
3. Orbital elements from new high-resolutionspectroscopy
For both CW Cep and U Oph, we obtained a new series of high-resolution échelle spectra using the High E ffi ciency and highResolution Mercator Échelle Spectrograph hermes on the 1.2 mMercator telescope at the Observatorio del Roque de los Mucha-chos, La Palma, Canary Islands, Spain. The hermes spectrographcovers the entire optical and NIR wavelength range (3700 - 9100Å) with a spectral resolution of R =
85 000 (Raskin et al. 2011).CW Cep was observed a total of 18 times over 13 nights. Threeobservations were taken in January 2015 with the remaining 15taken in August 2016. The argument of periastron progressed ∼ ◦ between these two subsets, and less than one degree withineither subset. U Oph, was observed 11 times over 10 nights fromApril to August 2016, during which time the argument of perias-tron progressed ∼ ◦ . The resulting spectra have an average S / Nof 110 in a range 51-144 and 145 in a range 117-163 for CW Cepand U Oph, respectively.
Article number, page 3 of 17 & A proofs: manuscript no. johnston_accepted_v1
Table 1.
Orbital parameters determined by method of spectral disentan-gling. The periods were fixed from photometry in these calculations.
Param. Unit CW Cep U Oph P d 2.72913159 1.67734590 T per d 57608.72 ± e - 0.0298 ± ω deg 218.7 ± K A km s − ± ± K B km s − ± ± q - 0.917 ± ± M A sin i M (cid:12) ± ± M B sin i M (cid:12) ± ± a sin i R (cid:12) ± ± hermes pipeline software package. This pipeline deliversmerged, un-normalised spectra. Therefore, before disentanglingthe spectra, we performed normalisation via spline function.Spectral disentangling (hereafter spd ) models the Dopplershift of spectral lines from a time-series of double-lined stellarspectra to determine the spectroscopic orbital elements as wellas simultaneously reconstruct the individual spectra of the com-ponents (Simon & Sturm 1994). Since the orbital elements aredirectly optimized in spd the determination of radial velocitiesfor each individual exposure is side-stepped. This removes thedependence on template spectra as are commonly used in thecross-correlation function (CCF) radial velocity (RV) determi-nation method, which is often a source of systematic error due tomismatches between the spectral type of the star and that of thetemplate (Hensberge & Pavlovski 2007). Moreover, the result-ing disentangled spectra of each component have an increasedsignal-to-noise compared to single-shot spectra, since disentan-gling acts as co-addition of the input spectra (c.f. Pavlovski &Hensberge 2010). To perform spd , we employ the FDB inary code (Ilijic et al. 2004), which performs spd in Fourier spacein order to e ffi ciently solve the large and over-determined sys-tem of linear equations represented by the data, through applyingdiscrete Fourier transforms to the spectra (Hadrava 1995).FDB inary calculates the RV curve for each component usingthe standard set of orbital elements: period P orb , time of perias-tron passage T per , eccentricity e , the argument of periastron ω ,and the semi-amplitudes of the RVs variations for the compo-nents K , and K . FDB inary simultaneously optimises all orbitalparameters across the entire set of spectra utilising the simplex algorithm. Although the Balmer lines dominate the optical spec-tra of hot stars, these lines are broad and usually cover a majorityof a single échelle order, thus, any imperfections in the order-merging and normalisation procedure would propagate into theoptimisation and a ff ect both the orbital elements and the result-ing disentangled component spectra. Therefore helium and metallines are more suitable for our purposes. The resulting optimisedorbital parameters for CW Cep and U Oph are listed in Table 1.The orbital parameters of CW Cep and U Oph have beenderived from fitting RVs in numerous previous studies. ForCW Cep, Stickland et al. (1992) determined K A = . ± . − , and K B = . ± . − ( q = . ± .
1) byfitting RVs extracted from three days of IUE spectra using aCCF method. However, since they had a relatively small num-ber of spectra (21), the authors chose to fix the eccentricityto e = . K A = ± − and K B = ± − ( q = . ± .
01) by fitting RVs obtained
Table 2.
Atmospheric parameters derived from an optimal fitting ofre-normalised disentangled spectra for the components of CW Cep andU Oph. For CW Cep a grid of NLTE synthetic spectra was used, whilstfor U Oph a grid of LTE synthetic spectra. The quantities given withoutthe uncertainties were fixed in the calculation.
Component T e ff log g ξ t v sin i [K] [dex] [km s − ] [km s − ]CW Cep A 28 300 ±
460 4.079 2.0 ± ± ±
420 4.102 1.5 ± ± ±
180 4.073 2.0 110 ±
6U Oph B 15 250 ±
100 4.131 2.0 108 ± K B derived by Popper & Hill (1991) is substantially larger thanthat obtained by Popper (1974) who used the very same data, butemployed the oscilloscopic method to determine RVs as opposedto the CCF method that was used by Popper & Hill (1991). Bycomparison, our results for CW Cep, listed in Table 1, place ourestimates within 1 σ of the solution presented by Stickland et al.(1992) and within 2 σ of Popper & Hill (1991).Popper & Hill (1991) also re-fit the orbital parameters ofU Oph on RVs determined via CCF from Lick Observatoryplate spectra, reporting K A = ± . − , and K B = ± − ( q = . ± . K A = ± − and K B = ± − ( q = . ± . ff erent esti-mates of K A = . ± . − and K B = . ± . − ( q = . ± .
01) from 34 plate spectra obtained by theESO 1.5m telescope. Until this work, the only results based onéchelle spectra were presented by Budding et al. (2009) whofound K A = . ± . − and K B = . ± . − ( q = . ± .
01) from 30 RV measurements determined via CCFfrom spectra obtained with the hercules spectrograph attachedto the 1m Canterbury University McLellan Telescope located atMt. John University Observatory in New Zealand. Our resultsare in rough agreement with the literature values, but highlightthe increased precision provided by spd which inherently min-imises uncertainties presented by line-blending and template-mismatches that plague CCF techniques.
4. Atmospheric parameters from disentangledspectra
CW Cep consists of two early-B spectral type stars with T e ff ∼
28 000 K (Popper 1974, 1980; Clausen & Gimenez 1991; Hanet al. 2002). These temperature estimates place both componentsin the temperature range where the strength of He ii lines startsto grow, thus allowing us to obtain precise e ff ective temperatureestimates through fine tuning the helium ionisation balance. Assuch, we apply the same methodology as described in Pavlovskiet al. (2018), which we briefly summarise here.As an observed spectrum of a binary is a composite of spec-tra of the two components, the disentangled spectra are equalto the intrinsic components’ spectra multiplied by the respec-tive light factors, i.e. the components’ fractional light contri-bution to the total light of a binary system, such that their co-addition reaches unity in the continuum. Generally, the frac-tional light contribution of each component can be determined Article number, page 4 of 17. Johnston et al.: Modelling of the B-type binaries CW Cep and U Oph
Table 3.
Abundances determined for the components of binary system CW Cep. The atmospheric parameters used for the calculation of modelatmospheres are given in Table 2. For the comparison the mean abundances for a sample of OB binaries given in Pavlovski et al. (2018), and for’present-day cosmic standard’ determined for a sample of a single sharp-lined B-type stars in Nieva & Przybilla (2012) are also presented.
Star C N O [N / C] [N / O] Mg SiCW Cep A 8.30 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Fig. 1.
Determination of the T e ff for the components of CW Cep (theprimary, component A, upper panels, the secondary. component B, bot-tom panels). The quality of fits are presented for He i λ ii λ i λ ii λ either in the light curve analysis, or extracted from disentangledspectra. In the case of partially eclipsing binary systems wherethe components have similar radii, the light ratios are degener-ate with the radii ratio and inclination. Therefore, it is advan-tageous to use the light ratio derived from disentangled spec-tra in the lightcurve modelling. We follow an iterative approach,where we first vary both the light factors and surface gravities,and then impose the light factors derived from spectroscopy aspriors in our lightcurve modelling. To obtain atmospheric param-eters, an optimised fit to the disentangled spectra of each compo-nent, which are re-normalised by their light-factor, is performedover a grid of pre-calculated non-local thermodynamic equilib-rium (NLTE) models using the starfit code (Tamajo et al. 2011;Kolbas et al. 2014). These theoretical NLTE spectra were cal- culated using A tlas detail/surface (Giddings 1981; Butler et al.1984). The synthetic spectra grid used in the optimisation con-tains models with T e ff ∈
15 000 −
32 000 K , and log g ∈ . − . / H] =
0. However, we are able tofix the log g for each component according to the values listed inTable 5, since high precision, independent estimates for the sur-face gravities were derived from the light curve modelling. Fix-ing the surface gravity e ff ectively lifts the degeneracy betweenthe e ff ective temperature and surface gravity, and enables us touse the Balmer lines as constraining information in our fit foundby the helium ionisation balance. By fixing the surface gravityand micro-turbulence per component, we reduce the optimisa-tion to eight free parameters: the e ff ective temperature T e ff percomponent, projected rotational velocity v sin i per component, arelative Doppler shift between disentangled spectra, and labora-tory reference frame, as well as the light-factors of the disentan-gled components. The optimisation across this parameter spaceis performed via a genetic algorithm modelled after that of thePIKAIA subroutine by Charbonneau (1995), with the errors cal-culated via Markov Chain Monte Carlo technique as describedby Ivezi´c et al. (2014) and implemented by Kolbas et al. (2014).The optimisation was carried out over the spectral segment from4000-4700 Å, and includes the Balmer lines H γ and H δ , in ad-dition to helium lines from both ionisation stages. Other spectrallines were masked. Due to the strong interstellar absorption bandwhich e ff ects the red wing of the H β line, we were unable to usethis spectral segment which covers the y filter. However, sincethe e ff ective temperatures of CW Cep A & B are similar, thewavelength dependence of the light-ratio is very small. The finalanalysis with fixed surface gravities and variable light ratios re-turned T e ff , p =
28 300 ±
460 K, and T e ff , s =
27 550 ±
420 K withlight-factors of 0 . ± .
005 and 0 . ± . i and He ii lines for both components is shownin Fig. 1.The reported values for the e ff ective temperature of the pri-mary of CW Cep have a broad range of almost 3 000 K, from T e ff , p =
28 000 ± T e ff , p =
25 400 in Terrell (1991), and with inter-mediate values T e ff , p =
26 500 K in Han et al. (2002) (Terrell(1991) and Han et al. (2002) fix T e ff , p and do not report formaluncertainties for these values). It should be noted, however, thatTerrell (1991) adopt their value for the primary e ff ective tem-perature from a spectral type classification of B0.5, and Clausen& Gimenez (1991) determine a mean value from di ff erent colorcalibrated photometric relations. In these studies, the e ff ectivetemperature of the secondary T e ff , s was then determined from thelight curve solution. The reported spread in secondary e ff ectivetemperature is only 1 300 K, with the hottest solution being only600 K (Clausen & Gimenez 1991) cooler than the primary and Article number, page 5 of 17 & A proofs: manuscript no. johnston_accepted_v1 the coolest solution being 1 900 K cooler than the primary (Ter-rell 1991). If we compare the values of our spectroscopically de-termined e ff ective temperatures for the components of CW Cep,and the di ff erence of their optimal values, T e ff , p =
28 300 ± ∆ T e ff = ±
620 K, to the various estimates in previ-ous analyses, we find the closest agreement with the estimatesof Clausen & Gimenez (1991). Comparatively, we are able toreduce the uncertainties considerably due to our methodologycombining the spectral disentangling, ionisation balancing, andfixing the surface gravity.Following our atmospheric analysis, we determine a detailedphotospheric composition for both stars as well. We calculateATLAS9 model atmospheres for the atmospheric parameters de-rived above, from which theoretical spectra are calculated withthe detail/surface suite. Details on the model atoms used can befound in Pavlovski et al. (2018). The abundances are then variedand optimised against the disentangled spectra, from which wereport abundances for carbon, nitrogen, oxygen, magnesium, andsilicon, as listed in Table 3. Additionally, we are able to derivethe microturbulence velocity ξ t from the oxygen-lines and thecondition of nul-dependence of the oxygen abundance on equiv-alent width. The derived ξ t values for CW Cep A and B are listedin Table 2. For comparison, the ’present-day cosmic standard’abundance pattern for sharp-lined early-B type stars of Nieva &Przybilla (2012) is provided in the bottom row of Table 3, withwhich we find general agreement. We also note that the abun-dances of CW Cep are in close agreement with the abundancepattern and ratios derived for OB binaries by Pavlovski et al.(2018) as listed in the third row of Table 3.Since iron lines are not visible in early-B type stars, the ironabundance can not be directly measured and used as a proxy forstellar metallicity. Instead, Lyubimkov et al. (2005) determinedthe magnesium abundance from the Mg ii line in a sample of 52un-evolved early- to mid-B type stars and used this as a proxy forstellar metallicity. Lyubimkov et al. (2005) determined the meanabundance log (cid:15) (Mg) = . ± .
15 to be in close agreement withthe solar magnesium abundance, log (cid:15) (cid:12) (Mg) = . ± .
02 as de-termined in Asplund et al. (2009). Exploiting the Mg abundanceas a proxy for metallicity, Lyubimkov et al. (2005) find that themetallicty of young MS B-type stars in the solar neighbourhoodand the Sun are the same. Following this work, we find that ourreported magnesium abundance suggests that CW Cep has solarmetallicity.Additionally, we note that we observe H α to be in emis-sion in the new spectra assembled for this work. Fig. 2 displaysspectra at roughly quarter phases as labelled, all of which showclear double-peaked emission with central absorption. The cor-responding velocity di ff erence between the blue ( V ) and red( R ) peaks remains constant at ∼
105 km s − through the orbitalphase. Similarly, we find that the intensity ration between the twopeaks remains roughly stable at V / R ∼ .
95 throughout the orbitas well. For comparison, in Fig. 2 we show also synthetic H α profile for 0.25 phase. Although H α emission is typical for Bestars or mass-transfer binaries we cannot reliably attribute theemission to a given component. Moreover, as there is no clearevidence of variability of the emission with the orbital phase, wepostulate that the emission originates from a circumbinary enve-lope or the nebula of the Cep OB3 association in which CW Cepis located. An extensive H i nebula in which the Cep OB3 as-sociation is embeded is well documented (e.g. Simonson & vanSomeren Greve 1976). Fig. 2.
Selected spectra corresponding to quarter phases centeredaround H α showing constant emission throughout the orbit. A syntheticcomposite spectrum of CW Cep at quarter phase is shown in red forcomparison. Radial velocity is calculated in the rest-frame of the sys-tem. U Oph consists of two main-sequence components of spectraltype (mid-)B. Given that our NLTE grid discussed in the pre-vious section is limited to stars hotter than 15 000 K, and thatthe use of the LTE formalism is overall justified for un-evolvedstars in this temperature range, we employ the Grid Search inStellar Parameters (GSSP, Tkachenko 2015) code for the anal-ysis of the disentangled spectra of the U Oph’s stellar compo-nents. The GSSP algorithm is based on a grid search in basicatmospheric parameters ( T e ff , log g , ξ , v sin i , and [M / H]) and, ifnecessary, individual atmospheric abundances, and utilizes a χ merit function and statistics to judge the goodness-of-fit betweenthe grid of synthetic spectra and the observed spectrum and tocompute 1 σ confidence intervals. Synthetic spectra are com-puted by means of the S ynth V radiative transfer code (Tsymbal1996) based on the pre-computed grid of LL models atmospheremodels (Shulyak et al. 2004). Both atmosphere models and syn-thetic spectra can be computed for arbitrary chemical composi-tions, where one, several, or all chemical elements’ abundancescan be set, also with an option of a vertical stratification in thestellar atmosphere. Similarly, the e ff ect of the microturbulent ve-locity can be taken into account, if necessary assuming its verti-cal stratification.GSSP is a multi-function software for spectrum analysis thatis able to deal with spectra of single stars (GSSP_ single mod-ule), and those of spectroscopic double-lined binaries, eitherwith their observed composite spectra (GSSP_ composite mod-ule) or with the disentangled spectra of individual stellar com-ponents (GSSP_ single or GSSP_ binary module). In the formerof the two binary cases (GSSP_ composite module), a compos-ite spectrum of a binary is fitted with a grid of composite syn-thetic spectra that are built from all possible combinations ofgrid points for the primary and secondary star. Individual radialvelocities can also be optimised along with all the aforemen-tioned atmospheric parameters of the two stars, where individ-ual flux contributions are taken into account by means of the Article number, page 6 of 17. Johnston et al.: Modelling of the B-type binaries CW Cep and U Oph stellar radii ratio factor. In the latter case, the distinction is madewhether the spectra are analysed as those of a single star with acertain light dilution factor (GSSP_ single module, so-called un-constrained fitting where the light dilution factor is assumed tobe independent of wavelength) or they are fitted simultaneouslyby optimising radii ratio to account for individual light contribu-tions (GSSP_ binary module, so-called constrained fitting withwavelength dependence of individual light contributions takeninto account). A simultaneous fit of the two disentangled spectrais essential when a binary consists of two stars which are signif-icantly di ff erent from each other in terms of their atmosphericproperties. In this instance, their relative light contributions willstrongly depend on wavelength. In the instance where the twostars have similar atmospheric parameters, independent fitting ofthe disentangled spectra is justified, while still enforcing that thetwo (wavelength-independent) light factors ultimately add-up tounity (see Tkachenko 2015, for detailed discussion).As with CW Cep, the atmospheric parameters of U Oph A &B are similar enough that we fit the disentangled spectra individ-ually. Again, we use an iterative approach where the light-ratiosare first determined from the disentangled spectra, then used aspriors in the light curve solution. The photometric surface gravi-ties are then fixed and the light-ratios are re-optimized along withthe other atmospheric parameters from the disentangled spectra.We found the light factors to be 0.575 ± ±
5. Revised Photometric Models
Both CW Cep and U Oph have been studied extensively in theliterature for several decades, with a heavy focus on the rapidapsidal motion displayed by the systems (Holmgren et al. 1991;Clausen & Gimenez 1991; Han et al. 2002; Wolf et al. 2002; Er-dem et al. 2004; Wolf et al. 2006; Vaz et al. 2007; Budding et al.2009). This study aims to use the updated mass ratio, semi-majoraxis and e ff ective temperatures of the primary and secondaryobtained in Sections 3 and 4 to determine updated dynamicalmasses, radii, and surface gravities from photometric modellingwith PHOEBE (Prša & Zwitter 2005; Prsa et al. 2011). For CW Cep we revisit the photometry initially analysed byClausen & Gimenez (1991). These data consists of 21 nightsof observations spanning 3 years in the Stromgren uvby photo-metric system, totalling 1396 measurements in the uby filters,and 1318 in the v filter. Both HD 218342 and HD 217035 servedas photometric comparison stars, from which the final di ff eren-tial magnitudes were obtained. Extinction corrections were ap-plied to the data as were determined by nightly coe ffi cients de-termined across the listed comparison stars and other standardobjects (Gimenez et al. 1990). According to Clausen & Gimenez(1991), the observations were constant to 0.004 mag in all filters,which we adopt as the uncertainty on each point.We also revisit archival data for U Oph, initially analysed byVaz et al. (2007). These data consist of 25 nights of observationsspanning 1992-1994 in the Stromgren uvby photometric system,totalling 645 measurements, however, due to a trend in the data,we do not use the u -band lightcurve. The data were taken withthe 0.5m ESO SAT telescope in La Sille, Chile. HR 6367, HR6353, and SAO 122251 were all used as comparison stars, fromwhich the final di ff erential magnitudes were obtained. As withCW Cep, extinction corrections were calculated each night from N o r m a li z e d f l u x N o r m a li z e d f l u x Fig. 3.
Quality of the fit of the disentangled (black dots) spectra with thesynthetic spectra (red and blue solid line for the primary and secondarycomponent, respectively) computed from the best fit parameters listedin Table 2. Spectra of the secondary component were vertically shiftedby a constant factor for clarity. the comparison stars used. For more information on the com-parison targets and observations, we refer to Vaz et al. (2007).Finally, Vaz et al. (2007) report a standard deviation of 0.0037mag in the vby filters, which we adopt as the uncertainty on eachpoint.
Both CW Cep and U Oph are well detached systems, exhibitingmild out of eclipse variability and slow apsidal motion, which forthe purposes of our modelling is e ff ectively mitigated by phase-binning the data. Our photometric modelling uses the PHOEBEbinary modelling code, which is a modern extension of the origi-nal WD code but also incorporates new physics such as dynamice ff ects, the light travel time e ff ect, and the reflection e ff ect (Prša& Zwitter 2005; Prsa et al. 2011). Given that all components con-sidered are expected to have radiative envelopes, we fix the grav-ity darkening exponent to unity for all components (von Zeipel Article number, page 7 of 17 & A proofs: manuscript no. johnston_accepted_v1 emcee a ffi ne-invariant en-semble sampler MCMC code (Foreman-Mackey et al. 2013),which has already been successfully applied by Schmid et al.(2015); Hambleton et al. (2016); Pablo et al. (2017); Johnstonet al. (2017); Kochukhov et al. (2018).MCMC procedures numerically evaluate Bayes’ Theorem,given by: p ( Θ | d ) ∝ L ( d | Θ ) p ( Θ ) , (1)to estimate the posterior probability, p ( Θ | d ), of some varied pa-rameters Θ given the data d .We can see above that p ( Θ | d ) isproportional to the product of the likelihood function L ( Θ | d )and the prior probability of the parameter vector p ( Θ ). We writethe likelihood function as:ln L ∝ − (cid:88) i (cid:32) d i − y ( Θ ) i σ i (cid:33) , (2)where, y ( Θ ) i is each individual model point and σ i are the indi-vidual uncertainties associated with the data. We have written thelog-likelihood function above as this is what is used in practice.To make e ffi cient use of the information obtained via the spec-troscopic analysis, we apply Gaussian priors on the light factors(per-cent contribution per component) and v sin i estimates percomponent, as well as the projected binary separation a sin i , themass ratio q , the e ff ective temperature of the secondary T e ff , ,and the eccentricity of the orbit. However, since PHOEBE doesnot directly sample all of these, we calculate the v sin i sepa-rately for each component and a sin i for every Θ considered.By including the spectroscopic light factors and simultaneouslyfitting all filters, we arrive at a more robust solution than if wewere to fit them all individually and mitigate any degeneraciesbetween the temperatures, light-factors, and potentials of eachcomponent (Clausen & Gimenez 1991). Furthermore, inclusionof priors on v sin i for each component helps constrain the spinparamters f = ω rot , /ω orb and f = ω rot , /ω orb , which are oth-erwise largely unconstrained.We draw parameter estimates and uncertainties as the me-dian and 68 .
27% (1 σ ) Highest Posterior Density (HPD) inter-vals of the marginalised posterior distribution for each sampledparameter. As both systems undergo apsidal motion, we bin eachlightcurve such that each phase bin covers 0.0033 phase units,which covers the entire periastron advance in a single binnedpoint for either system. Although PHOEBE accepts e and ω di-rectly, we sample e sin ω and e cos ω in our MCMC analysis andsolve for e and ω afterwards. To aid in the discussion and pro-vide additional constraints, we also report the relative radii in thebottom panel of Table 4. To propagate our newly derived spectroscopic and orbital infor-mation into updated dynamical masses and radii, we fix the ef-fective temperature of the primary (T e ff ) to the value listed inTable 2. As mentioned above, we apply Gaussian priors on themass ratio, the eccentricity, the projected binary separation, the v sin i per component, and the light-factor per component in the v - and b -band lightcurves, since these correspond to the spectralrange for which we derived the light factors. The light factor forthe u - and y -bands are given a uniform prior. For each sampled Θ , we interpolate limb-darkening coe ffi cients for the square-root law from the provided PHOEBE girds. Finally, given the radia-tive envelope of hot stars such as CW Cep A & B, we fix thealbedo to unity in both components.CW Cep is known to su ff er from third light which scales theapparent eclipse depths across each filter. Accounting for thisscaling is non-trivial as there is a degeneracy between inclina-tion and third light levels. However, this degeneracy is crucialto account for when determining the derived masses. We use auniform prior on the third light contributions per filter. Addi-tionally, we sample the reference date (HJD ), period ( P orb ), theinclination, the total binary separation, the secondary e ff ectivetemperature, as well as potentials and synchronicity parametersper component ( Ω , and f , , respectively), giving all uniformpriors. All sampled values, and their type of prior, are noted inTable 4.The analysis of Clausen & Gimenez (1991) states that theargument of periastron changes 24 ◦ from ∼ ◦ to ∼ ◦ overthe course of the photometric campaign, which corresponds tothe secondary minima shifting ∼ .
007 phase units. To mitigatethis change in periastron, we phase bin our data to 300 points,with each bin covering 0.0033 phase. Thus, the argument of pe-riastron that we sample does not correspond to the value pro-vided in the literature of the zero-point, but rather to the meanperiastron during the photometric campaign.The third column of Table 4 shows the median and HPD es-timates for the best fitting model. These values were used to con-struct the models seen in Fig. 4.For a consistency check, we compare the luminosities de-rived from the binary modelling with the luminosity derivedfrom the Gaia parallax for CW Cep: π G = . ± .
49 mas (Luriet al. 2018; Lindegren et al. 2018). We take A v = .
96 followingthe reported value of E( b − y ) from Clausen & Gimenez (1991)and BC v = . ± .
05 calculated as the average correction be-tween CW Cep A and B (Reed 1998). The summed luminos-ity derived from our binary model yields log LL (cid:12) = . ± . L G L (cid:12) = . ± . ∼
50% uncertainty on the Gaia parallax, wealso check the luminosity derived from the Hipparcos paral-lax ( π H = . ± .
69 mas; van Leeuwen 2007) which yieldslog L H L (cid:12) = . ± .
4. We find that all of these agree within theuncertainties.
As with CW Cep, we fix the e ff ective temperature of the primaryto the value listed in Table 2 and impose Gaussian priors on q , a sin i , e , v sin i per component and the light factors per compo-nent in the v - and b -band lightcurves. Although we can safelyignore the small eccentricity and set it to zero to perform spd ,we cannot ignore the eccentricity in the lightcurve. As such, weapply a Gaussian prior according to the values taken from Vazet al. (2007). Limb-darkening coe ffi cients are interpolated fromPHOEBE tables at every model evaluation. The albedos of bothcomponents are fixed to unity as both stars are expected to haveradiative envelopes.Since U Oph is also known to su ff er from third light, we takethe same approach as with CW Cep, using uniform priors for thethird light per filter and uniform priors in all other parameterslisted in Table 4. To mitigate the e ff ects of the apsidal advance,we phase bin into 300 bins, which e ff ectively covers the apparentchange in superior / inferior conjunction. Again, this means thatthe argument of periastron reported is an average over the pho-tometric campaign when the data was collected. The best model Article number, page 8 of 17. Johnston et al.: Modelling of the B-type binaries CW Cep and U Oph
Fig. 4.
Top panel displays the CW Cep PHOEBE Model (solid red) forthe Stromgren v lightcurve (black x’s) constructed from median valuesreported in Table 4. Lower panels show the residual lightcurves in the uvby filters after the best model has been removed. The dashed red-linedenotes the zero-point to guide the eye. according to the median estimates listed in Table 4 is shown inFig. 5. Derived parameters for both CW Cep and U Oph are re-ported in Table 5 alongside other solutions from the literature.As with CW Cep, we compare the total luminosity obtainfrom binary modelling with the luminosities derived from GAIA( π G = . ± .
13 mas; Luri et al. 2018; Lindegren et al. 2018)and Hipparcos ( π H = . ± .
41 mas van Leeuwen 2007), as-suming A v = . ± . LL (cid:12) = . ± .
01 does not agreewith the GAIA derived luminosity as log L G L (cid:12) = . ± .
09 within1 σ , but does agree with the Hipparcos derived luminosity aslog LL (cid:12) = . ± . σ .
6. Evolutionary Modelling
The updated masses, radii, and e ff ective temperatures ofCW Cep and U Oph provide strong discriminating power againststellar models. As discussed by Constantino & Bara ff e (2018)and Johnston et al. (2019), however, even such precision doesnot provide enough of a constraint to uniquely determine the ex-tent of the near-core mixing region. As such, we instead considerthe convective core mass, and treat the near-core mixing, param-eterised by a di ff usive exponentially decaying overshooting pre-scription with a scaled extent f ov , as a nuisance parameter. Todo this, we fit each component to a grid of isochrone-clouds asdescribed by Johnston et al. (2019). The isochrone-clouds areconstructed from MESA tracks computed at solar metallicity Z = . Y = .
276 (Nieva & Przybilla 2012),with α MLT = . τ , to cover the Fig. 5.
Top panel displays the U Oph PHOEBE Model (solid red) forthe Stromgren v lightcurve (black x’s) constructed from median valuesreported in Table 4. Lower panels show the residual lightcurves in the vby filters after the best model has been removed. The dashed red-linedenotes the zero-point to guide the eye. range f ov ∈ [0 . − .
04; 0 . ff usive expo-nential description of overshooting implemented in MESA.We adopt the Mahalanobis distance (MD) as our merit func-tion as applied in Johnston et al. (2019) and thorougly discussedin Aerts et al. (2018). Using the MD as our merit function al-lows us to account for correlations present amongst model pa-rameters that would otherwise compromise our solution (Aertset al. 2018; Johnston et al. 2019). We choose to fit the mass,adopting the errors listed in Table 5 instead of interpolating theisochrone-clouds to the dynamical values. Since the MD is amaximum-likelihood point estimator, we perform Monte Carlosimulations (with 10 000 iterations) to obtain confidence inter-vals on the model parameters and derived parameters of interest.We select the single best point returned for each iteration. Bykeeping only the best point, we sample the robustness of oursolution given our grid. If we were to keep the best N points,this would instead sample the variance of our solution space asa function of our grid, and observables, which, although is aninteresting phenomenon, is ultimately not the focus of this work.After 10 000 iterations, we bin the resulting distributions for allparameters of interest and apply 95% Highest Posterior Densityconfidence intervals. The results are listed in Table 6. The wide range of dynamical solutions for both CW Cep andU Oph shown in Table 5 gives reason for pause. The spread be-tween the minimum and maximum reported solutions is severaltimes larger than the formal uncertainties reported, despite thefact that the same photometric data-sets were used by di ff erentstudies. The main di ff erence across the individual solutions is themass ratio, or more fundamentally the spectroscopic data-sets. Article number, page 9 of 17 & A proofs: manuscript no. johnston_accepted_v1
Table 4.
Binary model parameters for CW Cep and U Oph.
CW Cep U OphParameter Prior HPD Estimate Prior HPD EstimateSampled ParametersL , u [%] U (40 ,
70) 57 . + . − . – –L , v [%] N (56 . , .
5) 56 . + . − . N (57 . , .
7) 57 . + . − . L , b [%] N (56 . , .
5) 56 . + . − . N (57 . , .
7) 57 . + . − . L , y [%] U (40 ,
70) 56 . + . − . U (40 ,
70) 57 . + . − . L , u [%] U (0 ,
15) 0 . + . − . – –L , v [%] U (0 ,
15) 1 . + . − . U (0 ,
15) 0 . + . − . L , b [%] U (0 ,
15) 2 . + . − . U (0 ,
15) 1 . + . − . L , y [%] U (0 ,
15) 3 . + . − . U (0 ,
15) 1 . + . − . T e ff , s [K] N (27550 , + − N (15620 , + − P orb [d] U (1 ,
5) 2 . + e − − e − U (1 ,
4) 1 . + e − − e − HJD [d] U ( − , + . + . − . U ( − , + . + . − . i [deg] U (70 ,
90) 81 . + . − . U (70 ,
90) 87 . + . − . e sin ω U ( − . , . − . + e − − e − U ( − . , . . + e − − e − e cos ω U ( − . , . . + e − − e − U ( − . , . . + e − − e − a [R (cid:12) ] U (5 ,
40) 24 . + . − . U (5 ,
40) 12 . + . − . q = M M N (0 . , . . + . − . N (0 . , .
01) 0 . + . − . Ω U (4 . ,
9) 5 . + . − . U (4 ,
9) 4 . + . − . Ω U (4 . ,
9) 5 . + . − . U (4 ,
9) 4 . + . − . f U (0 . ,
2) 1 . + . − . U (0 . ,
2) 1 . + . − . f U (0 . ,
2) 1 . + . − . U (0 . ,
2) 1 . + . − . Geometric Parameters r . + . − . . + . − . r . + . − . . + . − . Notes.
The top panel shows those parameters which were sampled during the MCMC run. For each parameter we list the units, when applicable,the priors, and the estimated values from the median and HPD confidence intervals. The bottom panel displays derived geometric parameters andtheir estimates. Gaussian priors are listed with an N , followed by their mean and width, and uniform priors are listed with a U , followed by theirboundaries. Table 5.
Derived Parameters CW Cep & U Oph.
CW CepParameter Gimenez et al. (1987) Clausen & Gimenez (1991) Han et al. (2002) a Han et al. (2002) b This WorkM [M (cid:12) ] 11 . ± . . ± .
14 13 .
49 12 .
93 13 . + . − . M [M (cid:12) ] 11 . ± . . ± .
14 12 .
05 11 .
84 11 . + . − . R [R (cid:12) ] 5 . ± . . ± .
12 6 .
03 5 .
97 5 . + . − . R [R (cid:12) ] 4 . ± . . ± .
12 4 .
60 4 .
56 5 . + . − . log g [dex] 4 . ± .
02 4 . ± .
02 4 .
01 3 .
99 4 . + . − . log g [dex] 4 . ± .
02 4 . ± .
02 4 .
19 4 .
19 4 . + . − . U OphParameter Holmgren et al. (1991) Vaz et al. (2007) Budding et al. (2009) This WorkM [M (cid:12) ] 4 . ± .
05 5 . ± .
091 5 . ± .
08 5 . + . − . M [M (cid:12) ] 4 . ± .
04 4 . ± .
072 4 . ± .
07 4 . + . − . R [R (cid:12) ] 3 . ± .
06 3 . ± .
020 3 . ± .
03 3 . + . − . R [R (cid:12) ] 3 . ± .
05 3 . ± .
034 3 . ± .
03 3 . + . − . log g [dex] 4 . ± .
01 4 . ± .
010 4 . ± .
01 4 . + . − . log g [dex] 4 . ± .
02 4 . ± .
012 4 . ± .
01 4 . + . − . Notes.
Table compares derived fundamental parameters from this work to previous studies of CW Cep (top) and U Oph (bottom). ( a ) Solution derived using spectroscopic values obtained by Popper & Hill (1991) ( b ) Solution derived using spectroscopic values obtained byStickland et al. (1992)Article number, page 10 of 17. Johnston et al.: Modelling of the B-type binaries CW Cep and U Oph
Table 6.
Monte Carlo Isochrone-cloud modelling 95% confidence in-tervals for CW Cep and U Oph.
Parameter CW Cep U OphAge [Myr] 7 . + − . + . − . f ov , . + . − . . + . − . f ov , . + . − . . + . − . M [M (cid:12) ] 13 . + . − . . + . − . M [M (cid:12) ] 12 . + . − . . + . − . X c , . + . − . . + . − . X c , . + . − . . + . − . M cc , [M (cid:12) ] 4 . + . − . . + . − . M cc , [M (cid:12) ] 3 . + . − . . + . − . Furthermore, each set of radial velocities used to calculate themass ratio was determined using di ff erent methods. Most criti-cally, this translates into a large disparity in estimated ages forthese systems, and therefore by necessity the estimated internalmixing. This is easily seen in the spread in ages for each systemdiscussed earlier in Section 2. In addition to di ff erent massesand radii being used, di ff erent e ff ective temperatures are also fitin the individual modelling e ff orts. In the end, these di ff erencese ff ectively mean that each study is modelling a di ff erent system.This highlights the need for a systematic evaluation of the ac-curacy versus the precision of dynamical and spectroscopic so-lutions for well studied eclipsing binaries. However, that is be-yond the scope of the work in this manuscript. We note that fu-ture studies which entail modelling e ff orts of samples comprisedof systems which were not homogeneously analysed must con-sider the systematic di ff erences between di ff erent methods. Wealso note the necessity in allowing the mass ratio, q , to vary. Inthe case where the mass ratio is fixed, the dynamical solutionreturns artificially high precision to the fourth or later decimalplace. Given the high-precision échelle spectra, combined withstate-of-the-art spd and MCMC methodologies, we find our so-lution to be more robust than previous solutions. As such, forthe remainder of the discussion, we only consider the results ob-tained in this work.As discussed previously, the evolutionary modelling ofeclipsing binaries involves several parameter degeneracies.While many studies attempt to constrain near core mixing, themodelling procedure is not directly sensitive to the details ofthe prescriptions of these phenomena, but rather to their conse-quences. As such, any inference drawn on stellar rotation, con-vective overshooting, and / or magnetism from evolutionary mod-elling is convoluted with additional e ff ects and uncertainties, atleast some of which can be attributed to the implementation ofsuch e ff ects as di ff usive processes in stellar structure and evolu-tion codes. Due to this, although we have shown that CW CepA and B and U Oph A and B are rotating at roughly a quar-ter of their critical rotation rates, any internal mixing caused bythis will be degenerate with mixing caused by convective over-shooting. Therefore, reflecting this and the discussions presentedby Constantino & Bara ff e (2018) and Johnston et al. (2019), wegear our discussion towards the core properties, which evolution-ary modelling is more directly sensitive to since these propertiesdictate the stellar evolutionary sequence.Despite the per-cent level precision on dynamical quantitiesprovided by the binary solution, our modelling could not providea constrained range for the extent of near-core mixing for the primary of CW Cep or for either component in U Oph. However,our modelling shows that CW Cep B requires a large amount ofinternal mixing to have its current observed properties and beco-evolutionary with CW Cep A. Stated di ff erently, CW Cep Brequires a more massive core than models based solely on theSchwarzchild criterion, otherwise it would appear as a di ff erentage compared to CW Cep A. The left panels of Figures 6 and 7show how the isochrone-clouds of ages reported in Table 6 coverlarge, and often overlapping, regions of the spectroscopic param-eter space due to the spread in near-core mixing. As can be seenin the accompanying right panels of said figures, this translatesto a generally more confined region in core properties shown inblack circles and black x’s for the primary and secondary, respec-tively for either system. At the age and mass range for CW Cep,the components have not progressed su ffi ciently through theirMS lifetimes to be able to critically constrain their core proper-ties, with the primary and secondary being ∼
27% and ∼ ∼ + − % and ∼ + − % of the total mass, respec-tively.As for U Oph, the primary and secondary are ∼
40% and ∼
31% through their respective MS lifetimes. In this case, the re-sulting core parameter regions are more constrained. The coresof U Oph A and B contain ∼ + − % and ∼ + − . % of the totalmass, respectively.The age estimate we find for CW Cep largely agrees with theestimates of Jordi et al. (1996) and Clausen & Gimenez (1991),but is nearly twice as old as the solution reported by Ribas et al.(2000b). Our age estimate for U Oph is considerably higher thanthe median reported value in the literature, but does agree withthe upper limits of those solutions reported with a lower metallic-ity closer to the value we find and use in our modelling. We notethat in both cases, the estimated ages agree with those reportedby previous studies. However, the solutions presented here aresystematically older than those presented by Schneider et al.(2014) for both CW Cep and U Oph. Furthermore, our solutionfor CW Cep show the components as much less progressed alongthe MS ( ∼
27% and ∼ ∼ −
40% and ∼ − ∼
40% and ∼
31% progress that we report. How-ever, Schneider et al. (2014) use a solution which is ∼
5% moremassive compared to ours. This again highlights the need for ahomogeneously analysed sample to draw inference on trends incore properties and the physical processes which influence them.
According to our MCMC estimates, CW Cep A and U Oph Bare both rotating super-synchronously by 2 σ , while CW CepB and U Oph A are both rotating synchronously within uncer-tainties. By combining the synchronicity parameters f , v A = ± − and v B = ± − , respectively, and U Oph A and B are rotatingat v A = ± − and v B = ± − , respectively.We also calculate the critical rotation rates for each star usingthe parameters listed in Table 5. We find the critical rotation ratesfor CW Cep A and B to be v crit , A = ± − and v crit , B = Article number, page 11 of 17 & A proofs: manuscript no. johnston_accepted_v1 ± − , respectively. This reveals that CW Cep A and Bare rotating at 19 . ± .
6% and 17 . ± .
6% their critical rates,respectively. For U Oph A and B, we find v crit , A = ± − and v crit , B = ± − , respectively. Thus revealing themto be rotating at 25 . ±
2% and 24 . ±
1% their critical rates,respectively.Tidal theory gives predictions of synchronisation and circu-larisation timescales for binary systems (Zahn 1975, 1977). Us-ing the results of our modelling, we find τ sync = . ± . τ circ = . ± τ sync = . ± . τ circ = . ± .
02 Myr for U Oph. The agesobtained in our isochrone-cloud modelling are an order of mag-nitude larger than the theoretical synchronisation timescales foreither CW Cep or U Oph. CW Cep is significantly younger thanits theoretical circularisation timescale, and given the masses ofthe components, it will evolve beyond the MS before circularisa-tion occurs. U Oph, however, is already significantly older thanits circularisation timescale, but is still observed to be eccen-tric. This observation fits with the presence of a third body thatlikely sends the system through Kozai-Lidov cycles, as opposedto having a constantly decaying eccentricity.
7. Conclusions
Contemporary binary modelling techniques have the ability toprovide per-cent level (or better) precision on fundamental stellarparameter estimates to be compared against evolutionary mod-els. These parameter estimates have been used by numerousstudies, including this one, to attempt to constrain poorly under-stood near-core mixing processes which cause deviations fromnominal stellar evolution. However, no clear consensus exists inthe literature as to wether or not this is possible.In this work we obtained and analysed new spectroscopicobservations on the intermediate- to high-mass binaries CW Cepand U Oph. Our analysis yielded an updated mass-ratio to beused for lightcurve modelling, as well as the first abundance pat-terns for these systems. The abundance patterns were revealedto be roughly solar, which was exploited in the isochrone-cloudevolutionary modelling.We performed lightcurve modelling using a BayesianMCMC optimization routine wrapped around the PHOEBE bi-nary modelling code to obtain updated and highly precise massand radius estimates. These estimates roughly agree with paststudies, but the spread in reported solutions is much larger thanthe precision reported for any solution. This raises a concern inrelation to the robustness of the accuracy of a solution versusits precision. To test the consistency of our solutions, we com-pared the luminosities from binary modelling with those calcu-lated from Gaia parallaxes. Furthermore, given the close sepa-ration of the components in these systems, they are ideal candi-dates for investigating the influence of the inclusion of secondorder physics such as Doppler boosting and reflection on result-ing modelled core properties. However, such analysis requireshigh-precision space photometry which has yet to be assembledfor these systems, but will be done soon by the
TESS mission(Ricker et al. 2015). Additionally, the high precision orbital anddynamical solutions allowed us to investigate the rotation ratesand tidal synchronisation and circularisation timescales for bothsystems.Using our updated dynamical solution and temperatures, weperformed isochrone-cloud modelling following the procedureas described by Johnston et al. (2019) to obtain estimates onboth model input parameters, as well as derived parameters such as the core properties. Our results reveal that, given model de-generacies, we cannot critically constrain the extent of near-coremixing. We do, however, constrain the core mass and hydrogencontent for both components of CW Cep and U Oph, as thesequantities directly dictate the current evolutionary status of a star.We compare our results to those of Schneider et al. (2014), whoperformed a similar analysis, but which assumed rotational mix-ing instead of exponentially decaying di ff usive convetive over-shooting in their evolutionary models. Combined with the alarm-ing spread in reported dynamical solutions shown in Table 5,our comparison highlights the need for a homogeneously anal-ysed sample to be able to make meaningful inference on internalphysical processes such as convective overshooting, rotationaland pulsational mixing, and magnetism (Aerts et al. 2014).Finally, we ask that future studies which perform evolution-ary modelling report core masses of their solutions in addition tothe overshooting extent or near-core rotation rate. Acknowledgements
We thank the referee for their helpful comments which haveimproved the manuscript. We thank Dr. Dominic Bowman andProf. Hugues Sana for their discussions pertaining to error de-termination. The research leading to these results has receivedfunding from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation pro-gramme (grant agreement N ◦ / / ffi ce (BELSPO) throughPRODEX grant PLATO (AT). KP acknowledges financial sup-port from the Croatian Science Foundation under grant IP-2014-09-8656 (STARDUST). The computational resources and ser-vices used in this work were provided by the VSC (Flemish Su-percomputer Center), funded by the Research Foundation - Flan-ders (FWO) and the Flemish Government – department EWI.Based on observations made with the Mercator Telescope, op-erated on the island of La Palma by the Flemmish Community,at the Spanish Observatorio del Roque de los Muchachos of theInstituto de Astrofísica de Canarias. References
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Fig. 6. Left : Isochrone-clouds for the ages reported in Table 6 with the spectroscopic uncertainties plotted for CW Cep A and B in black.
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Article number, page 15 of 17 & A proofs: manuscript no. johnston_accepted_v1
Fig. A.1.
Marginalised Posterior Distributions for primary and sec-ondary passband luminosities for each observed filter. Median denotedby solid vertical red line, upper and lower bounds for 68.27% CI de-noted by dashed vertical red lines.
Fig. A.2.
Marginalised Posterior Distributions for primary and sec-ondary parameters. Median denoted by solid vertical red line, upper andlower bounds for 68.27% CI denoted by dashed vertical red lines.
Fig. A.3.
Marginalised Posterior Distributions for orbital parameters.Median denoted by solid vertical red line, upper and lower bounds for68.27% CI denoted by dashed vertical red lines.
Fig. A.4.
Marginalised Posterior Distributions for system parameters.Median denoted by solid vertical red line, upper and lower bounds for68.27% CI denoted by dashed vertical red lines.
Appendix A: CW Cep Marginalised posteriordistributionsAppendix B: U Oph Marginalised posteriordistributions
Article number, page 16 of 17. Johnston et al.: Modelling of the B-type binaries CW Cep and U Oph
Fig. B.1.
Marginalised Posterior Distributions for primary and sec-ondary passband luminosities for each observed filter. Median denotedby solid vertical red line, upper and lower bounds for 68.27% CI de-noted by dashed vertical red lines.
Fig. B.2.
Marginalised Posterior Distributions for primary and sec-ondary parameters. Median denoted by solid vertical red line, upper andlower bounds for 68.27% CI denoted by dashed vertical red lines.
Fig. B.3.
Marginalised Posterior Distributions for orbital parameters.Median denoted by solid vertical red line, upper and lower bounds for68.27% CI denoted by dashed vertical red lines.